Build an arch of wedge-shaped stones (voussoirs) whose joints all point towards the centre of the curve: each stone can then fall only by forcing its neighbours apart, and so the ring holds itself up once the last stone, the keystone, is in place.
joints ⟂ intrados (radial to the centre)
An arch pushes outward on its supports as well as down. For the same load and span the horizontal push grows as the rise falls: halve the rise and you roughly double the thrust.
H ≈ w·s² ⁄ 8f (uniform load w per metre, span s, rise f)
Trace the path of the resultant force through the stones from crown to foundations. If a line can be found that stays everywhere within the masonry, the arch will stand; where it touches an edge, a hinge will open, and four hinges make it collapse.
Draw a pointed arch from two centres on the springing line. With the centres at the springings, the arch is equilateral; with centres beyond them, a lancet; with centres between them, a drop arch. Late Gothic added arches of four centres and the reversed curve of the ogee.
r = s (equilateral) · r > s (lancet) · s⁄2 < r < s (drop)
Classify arches by the geometry of the intrados: the number and position of the centres (one for round and segmental, two for pointed, three or four for basket and Tudor arches), or a single continuous curve (ellipse, parabola, catenary), and distinguish true arches, with voussoirs, from corbelled “arches” of oversailing courses.
A round arch must rise half its span, so arches of different spans on one springing line have crowns at different heights. A pointed arch can be given any rise at least half its span, so arches of different widths can all be made to reach the same height.
round: f = s⁄2 (fixed) · pointed: r = (s²⁄4 + f²) ⁄ s for any f ≥ s⁄2
An arch cannot support itself until it is complete. Build it on a timber centring shaped to the intrados, strike the centring only when the keystone is set and the mortar has taken, and reuse it from bay to bay, which is one reason to keep the spans of an arcade equal.
In a row of arches the thrusts of neighbouring arches cancel over each intermediate pier, which may therefore be slender; at the ends of the row nothing balances them, so the outermost piers must be made wider or buttressed.
Over a doorway or window spanned by a lintel, turn an arch in the wall above so that the weight of the masonry is carried round the opening onto the jambs, leaving the lintel to carry only the small triangle of wall beneath the arch.
Raise the central nave above the lean-to roofs of the aisles and pierce the upper walls with a row of windows, the clerestory, so that the middle of a wide hall is lit from high on both sides.
nave height ≈ 1½–2 × nave width; one window over each bay of the arcade
Cover a wide nave with triangulated timber trusses rather than a vault: their tie-beams take the outward thrust, so the walls need only carry weight and may stand on slender columns.
nave span ≈ 13–24 m on trusses (Santa Sabina ≈ 13.5 m; Old St Peter’s ≈ 24 m)
Order the church as a single path: a colonnaded atrium, a narthex, the long nave drawn toward the altar, and the apse framed by a triumphal arch at the end of the axis.
Throw an arch diagonally across each upper corner of the square, backed by a small conch or cone; the four squinches and the four walls between them make an octagon, on which a round dome can sit.
octagon side = S (√2 − 1) ≈ 0.414 S; squinch span = octagon side
Design a rose by dividing the circle into equal sectors; fill each with the same traceried unit (a light with a cusped pointed head, an oculus above) radiating from a central hub, so that the whole is generated from one sector by rotation.
n petals · each sector 360° ⁄ n
End the choir in a semicircle of piers; wrap an aisle (the ambulatory) around it, and open a ring of chapels off the ambulatory, each with its own altar and windows, so that the east end steps down in tiers of roofs and buttresses.
Lay out the church as an Early Christian basilica (a tall nave lit from above, lower aisles on either side, a sanctuary ending in an apse to the east) and cross it with a transept, so that the plan becomes a Latin cross with a tower over the crossing.
Carry vaults on walls thick enough to absorb their thrust within their own mass; keep the windows small and set them low, below the springing of the vault, and strengthen the walls outside with shallow pilaster buttresses.
walls commonly 1½–2 m thick; buttresses shallow, window openings narrow and splayed
Let the square of the crossing set the measure of the whole church: the nave and transept arms are multiples of it, each nave bay is a square of the same size, and each aisle bay a square one quarter its area.
nave bay = crossing square; aisle bay = ¼ of it; nave length = n squares
Where one nave vault spans two bays of the arcade, give the principal piers, which receive the vault, a heavier compound section with shafts running up the wall, and make the intermediate supports lighter: columns or simpler piers.
A–B–A (pier, column, pier) · or A–B–B–A in Saxon churches
Run a shaft or pilaster from the floor, up the face of the pier and the wall above, to the roof or vault, so that the long nave wall is divided into a sequence of vertical bays, each containing one arch of the arcade and the windows above it.
Strengthen a barrel-vaulted nave with transverse arches at every bay, springing from shafts on the piers; they stiffen the vault, mark the bays and serve as ribs on which the vault can be built bay by bay.
Build galleries (tribunes) over the aisles and cover them with quarter-circle (quadrant) vaults that lean against the nave wall just below the springing of its barrel, so that the gallery vaults carry the barrel’s thrust down to the outer walls.
Multiply altars at the east end by opening apses off the transept arms and the choir, stepped back from the main apse in echelon, each with its own small semidome and roof.
Articulate plain rubble walls outside with shallow vertical strips (lesenes) linked at the top by a frieze of small round arches on corbels: the Lombard band.
Mark the west end with a massive, towered block containing an entrance hall below and a chapel or gallery above, rising higher than the nave: a church in front of the church.
Pitch a roof no lower than its covering allows (roughly 45° for thatch, 35° for plain tiles, 20–25° for slate, 15–20° for interlocking tiles, a few degrees for sheet metal) and steeper where snow lies long or rain is heavy.
thatch ≥ 45° · plain tile ≥ 35° · slate ≥ 22½° · pantile ≥ 17½° · metal ≥ 5° · membrane ≈ 1–2°
To roof a wide span, join each pair of rafters to a tie beam at their feet so that the three form a triangle that cannot spread; hang the tie beam from the apex by a king post, or by two queen posts on larger spans.
Build the baptistery, and its font, with eight sides, because the eighth day, the day of the resurrection, is the day into which the baptised are born.
plan and font = octagon (8 = 7 + 1, the day after the Sabbath)
Build the monastery, if possible, so that everything it needs (water, a mill, a garden and the various crafts) is inside the enclosure, so that the monks need not wander outside, which does their souls no good; and set a wise old monk at the gate, with a cell beside it.
Let all the monks sleep in one place if their number allows, each in his own bed, clothed and ready to rise, with a light burning until morning; place the dormitory where they can go straight to the church for the night office.
Receive every guest as Christ, the poor and pilgrims with the greatest care; give the abbot and the guests a kitchen of their own, so that guests arriving at unexpected hours do not disturb the brothers; lodge them in a guest house apart, in the care of a God-fearing brother.
Give the sick a room set apart and a brother to serve them, with baths when they need them; give the novices a place of their own where they study, eat and sleep under a senior.
Put the church in the centre with its altar to the east; the cloister against its south side, ringed by dormitory, refectory and cellar; the abbot, the school and the guests to the north; the novices, the sick and the gardens to the east; the kitchens, workshops, mills, barns and animals to the south and west; the hospice for the poor by the entrance.
A barrel (tunnel) vault is an arch extruded along its axis. It thrusts outward all along both sides, so it must rest on continuous thick walls, which can be pierced only sparingly, and only below its springing.
thrust per metre of wall ≈ constant along the whole length
Intersect two barrel vaults of equal height at right angles: the result is a groin vault, whose weight and thrust are gathered along the diagonal groins to the four corners, leaving the sides free for openings.
surface = max(barrel₁, barrel₂) · groins on the diagonals
Where the groin vault is the union of two barrels, the cloister (domical) vault is their intersection: four curved cells rise from all four walls and meet in valleys along the diagonals. It needs walls on every side.
surface = min(barrel₁, barrel₂) · valleys on the diagonals
Two semicircular barrels of different spans rise to different heights. To groin-vault an oblong bay the narrower barrel must be stilted or flattened, and the groins then wander out of plane: the difficulty that pushed builders towards ribs and pointed arches.
Build the arches of the vault first (transverse, wall and diagonal ribs) on light centring, then fill the triangular cells between them with thin webs of stone laid course by course from rib to rib.
Raise the crown of each bay well above the tops of its boundary arches, so that the vault becomes a shallow dome on ribs whose thrust falls more steeply onto the walls.
crown ≈ half-diagonal above the springing; boundary arches ≈ half-span